Error estimates of finite element method for semi-linear stochastic strongly damped wave equation

TL;DR

This paper analyzes the error estimates of finite element methods applied to a semi-linear stochastic strongly damped wave equation driven by Gaussian noise, establishing optimal convergence rates based on solution regularity.

Contribution

It provides the first comprehensive error analysis for finite element discretizations of this specific stochastic damped wave equation, including regularity and convergence results.

Findings

Optimal convergence rates in space and time

Regularity of mild solutions with space-time white noise

Numerical examples confirming theoretical error estimates

Abstract

In this paper, we consider a semi-linear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework, we establish existence, uniqueness and space-time regularity of a mild solution to such equation. Unlike the usual stochastic wave equation without damping, the underlying problem with space-time white noise (Q = I) allows for a mild solution with a positive order of regularity in multiple spatial dimensions. Further, we analyze a spatio-temporal discretization of the problem, performed by a standard finite element method in space and a well-known linear implicit Euler scheme in time. The analysis of the approximation error forces us to significantly enrich existing error estimates of semidiscrete and fully discrete finite element methods for the corresponding linear deterministic equation. The main results show optimal convergence rates in the sense that the orders of convergence in space and in time coincide with the orders of the spatial and temporal regularity of the mild solution, respectively. Numerical examples are finally included to confirm our theoretical findings.